From the proof of the above Lemma in the case \(m=0\)  which was established in [1]} we have the following estimate:
\(\begin{split}\int _{S_{\tau _2}} & J^N [ N^{m+1} \psi _{\ge \ell } ] \cdot n_{\tau _2} \, d\mu _{S_{\tau _2}} + \int _{\tau _1}^{\tau _2} \int _{S_{\tau }} J^N [ N^{m+1} \psi _{\ge \ell } ] \cdot n_{\tau } \, d\mu _{S_{\tau }} d\tau \\ \le & C \int _{S_{\tau _1}} J^N [ N^{m+1} \psi _{\ge \ell } ] \cdot n_{\tau _1} \, d\mu _{S_{\tau _1}} + \int _{S_{\tau _1}} J^N [ N^{m+1} T \psi _{\ge \ell } ] \cdot n_{\tau _1} \, d\mu _{S_{\tau _1}}\\ & + C\int _{\tau _1}^{\tau _2} \int _{S_{\tau } \cap \lbrace r_1 \le r \le r_2 \rbrace } J^T [ N^{m+1} \psi _{\ge \ell } ] \cdot n_{\tau } \, d\mu _{S_{\tau }} d\tau + C\int _{\tau _1}^{\tau _2} \int _{S_{\tau }} J^T [ N^{m} T \psi _{\ge \ell } ] \cdot n_{\tau } \, d\mu _{S_{\tau }} d\tau \\ & + C\int _{\tau _1}^{\tau _2} \int _{S_{\tau } \cap \lbrace  r \le r_2 \rbrace } J^T [ N^{m} \psi _{\ge \ell } ] \cdot n_{\tau } \, d\mu _{S_{\tau }} d\tau + C\int _{\tau _1}^{\tau _2} \int _{S_{\tau }} | G_{m, \ell } |^2 \, d\mu _{S_{\tau }} d\tau ,\end{split}\) 
